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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">m</font></span><br/>

<span class="kw">thus </span><a NAME="E1:20"/>
 <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">m</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> $1;<br/>
<a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">k</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 <span class="kw">and </span><br/><a NAME="E2:20_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]</span><br/>

<a NAME="E3:20_1_1"/>
<span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T13" target="_self" title="FIB_NUM3:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T14" target="_self" title="FIB_NUM3:th.14">Th14</a>, <a class="ref" href="fib_num3.html#T15" target="_self" title="FIB_NUM3:th.15">Th15</a></span>;<br/>
<a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T11" target="_self" title="FIB_NUM3:th.11">Th11</a>, <a class="ref" href="fib_num3.html#T14" target="_self" title="FIB_NUM3:th.14">Th14</a></span>;<br/>
<a NAME="E4:20_1"/>
 for <font color="Olive" title="b1">k</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fib_num2.html#S1" title="FIB_NUM2:sch.1">FIB_NUM2:sch 1</a>(<a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:20_1"/>
 <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


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